Deriving Expected Shortfall as an Unconditional Expectation

Summary

The note derives an unconditional-expectation form of expected shortfall from its definition as the negative conditional mean of returns in the loss tail. It uses the indicator of the event that a return falls below the VaR threshold, then applies the conditional-expectation identity: the expectation of a variable restricted to an event is divided by that event’s probability. When the VaR threshold defines a tail event with probability α, this gives the indicator-weighted return divided by α, with a negative sign.

The division by α normalizes the tail contribution by the probability of entering that tail, recovering the average loss conditional on a tail event. This formulation is relevant to Acerbi’s second expected-shortfall backtest, which motivates the question. The derivation relies on the tail event having probability α under the VaR definition used. Discrete distributions, probability mass at the threshold, or differing quantile conventions can complicate that equality, so the event probability may not equal α exactly in every setting.

Key ideas

• Conditional expectation on an event equals the indicator-weighted expectation divided by the event probability.
• Expected shortfall is the negative mean return conditional on a return falling beyond the VaR threshold.
• If the VaR tail event has probability α, dividing by α converts its unconditional contribution into a conditional tail average.
• The equality may require care when the distribution has mass at the VaR threshold.

Tags

# expected shortfall as unconditional expectation


# expected shortfall as unconditional expectation












Acerbi has several backtests for expected shortfall. The second backtest is based on this equality

Does anybody know how to derive this equality? Can anybody explain, why it makes sense, especially dividing by $\alpha$?

Background: I took the equality from this presentation https://www.cass.city.ac.uk/faculty-and-research/faculties/finance/seminars-and-workshops/financial-engineering-workshops/ACERBI-Carlo-10.03.2015.pdf

This is how I know expected shortfall

## Answer by Daneel Olivaw (score 3, accepted)

https://quant.stackexchange.com/a/35072

Letting $X_t$ be a random variable, its conditional expectation with respect to some event $E$ is given by:

$$ \mathbb{E}^{\mathbb{P}}[X_t|E]=\frac{\mathbb{E}^{\mathbb{P}}[\mathbf{1}_EX_t]}{\mathbb{P}(E)}$$

In our case, Expected Shortfall is defined as:

$$ \text{ES}_{\alpha,t} = -\mathbb{E}^{\mathbb{P}}[X_t|X_t + \text{VaR}_{\alpha} < 0]$$

Hence:

$$ \begin{align} \text{ES}_{\alpha,t} & = -\frac{\mathbb{E}^{\mathbb{P}}[\mathbf{1}_{\{X_t + \text{VaR}_{\alpha} < 0\}}X_t]}{\mathbb{P}(X_t + \text{VaR}_{\alpha} < 0)} \\[9pt] & = -\frac{\mathbb{E}^{\mathbb{P}}[\mathbf{1}_{\{X_t + \text{VaR}_{\alpha} < 0\}}X_t]}{\alpha} \\[11pt] & = -\mathbb{E}^{\mathbb{P}}\left[\frac{\mathbf{1}_{\{X_t + \text{VaR}_{\alpha} < 0\}}X_t}{\alpha}\right] \end{align} $$

Where the second step is a consequence of the definition of $\text{VaR}_{\alpha}.$

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.